__Microscopic view of Ohm's law__

The motion of conduction electrons in an electric field (E) is the combination of the motion due to random collisions and E. When we consider all the free electrons, their random motions average to zero and make no contribution to the drift speed. Thus, the drift speed is due only to the effect of the electric field on the electrons.

We know that

v= u + aT, where v= drift velocity, u=initial velocity, and T=time between each collision.

And since u= 0 after each collision, we can write the equation as

We know that

v= u + aT, where v= drift velocity, u=initial velocity, and T=time between each collision.

And since u= 0 after each collision, we can write the equation as

we can relate (a) to the electric field (E) in the wire by using newtons equation

F= m*a where m is the mass of an electron.

F= m*a where m is the mass of an electron.

From equation (i) and (ii), we will get

we will get

which can be written as,

we will see that E and J are the same except P (rho), therefore, we can write the equation for P (rho) as